On an Auxiliary Nonlinear Boundary Value Problem in the Ginzburg-Landau Theory of Superconductivity and its Multiple Solutions

We realize analytic-numerical investigation of a homogeneous boundary value problem (BVP) for a second-order ordinary differential equation (ODE) with cubic nonlinearity and two real parameters which arises from the Ginzburg-Landau theory of superconductivity. Multiple nontrivial solutions to this problem depending on the specified parameters are expressed through the Jacobi elliptic functions and describe the stationary states (near the critical values of temperature) of a superconducting infinite plate of a finite thickness without magnetic field. It is a “degenerate” problem with respect to the original nonlinear BVP for a superconducting plate in a magnetic field and is important to construct algorithm for finding all the solutions to the indicated input problem in a wide range of the parameters. Studied problem is of separate mathematical interest by itself.

The Ginzburg-Landau theory of superconductivity, stationary states of a superconducting plate without magnetic field, nonlinear second-order ordinary differential equation (ODE), homogeneous boundary value problem (BVP) and its multiple solutions